• e(t)

D J(t) ~(t) I

L n np Q R

u V a Ii 0 (J A X" E pee)

concentration emerging from batch reactor after residence time t

coefficient of longitudinal diffusion residence time distribution frequency function integral residence time distribution number of passages through recycle reactor

= length of longitudinal diffusion reactor = number of CSTR's in series = integer above Xp(R + 1) / (J = volumetric flow rate of feed = ratio of recycle flow to feed flow = time

axial velocity in longitudinal diffusion reactor reactor volume age of fluid element in reactor mean age delta function mean residence time V / Q life expectation of fluid element in reactor mean value of X for a point . mean age of fluid elements with life expectation A

= rate of chemical reaction at concentration c

CHUE-PL 67-0590 0'2 = variance f(X) = life expectation distribution frequency function var a = variance of age of fluid elements in reactor var all = variance of age between points var a1 = variance of age within points

Literature Cited

Asbj(4rnsen, O. A., A.I.Ch.E.-1. Chern. E. Joint Meeting, London, June 1965, Symposium on Mixing, Paper 10.5a.

Danekwerts, P. V., Chem. Eng. Sci. 8,93 (1958). Douglas, J. M ., Chem. Eng. Progr. Symp. Ser. 60, No. 48, 1 (1964). Gillespie, B. M., Carberry, J. J., Chem. Eng. Sci. 21, 472 (1966a). Gillespie B. M ., Carberry, J. J ., IND. ENG. CHEM. FUNDAMENTALS

5,164 (1966b). , Ng, D. Y. C., Rippin, D. W. T., Proeeedings of Third European

Symposium on Chemical Reaction Engineering, Amsterdam, September 1964, p. 161, Pergamon Press, New York, 1964.

Van de Vusse, J. G., Chem. Eng. Sci. 21, 611 (1966). Weinstein, H., Adler, R. J., Chem. Eng. Sci. 22, 65 (1967). Zwietering, Th. N., Chern. Eng. Sci. 11, 1 (1959).

RECEIVED for review August 15, 1966 ACCEPTED May 10, 1967

VAPOR·LIQUID EQUILIBRIA AT HIGH

PRESSURES. VAPOR-PHASE FUGACITY

COEFFICIENTS IN NONPOLAR AND

QUANTUM-GAS MIXTURES • P. L. CHUEH AND J. M. PRAUSNITZ

Department of 'Chemical Engineering, University of Caltfornia, Berkeley, Calif.

Vapor-phase fugacity coefficients, calculated with the Redlich-Kwong equation, are in excellent agreement

with experimental results when Redlich's original mixing rule for constant a is modified by inclusion of one

binary interaction constant. Such constants are reported here for 115 binary systems including several

which contain hydrogen, neon, or helium. Quantum corrections for these gases are taken into consideration.

The calculations are readily performed with an electronic computer and are useful for reduction and correla­

tion of vapor-liquid equilibrium data at high pressures.

THERMODYNAMIC reduction and correlation of vapor-liquid equilibrium data are common at low pressures, but de­

spite a generous supply of equilibrium data at high pressures, little attempt has been made to reduce such data with thermo­dynamically significant functions .

In high-pressure phase equilibria it is not possible to make various simplifying assumptions commonly made at low pressures and, as a result, thermodynamic analysis has only rarely been applied to high-pressure systems. In such systems, both phases, vapor and liquid, exhibit large deviations from ideal behavior. In previous papers we considered the liquid phase (Chueh et al., 1965, 1967B) but in this work we consider the vapor phase; we are concerned with a reliable technique for calculating vapor-phase fugacity coefficients in nonpolar mixtures, including those containing one of the quantum gases. Toward that end we propose to use the Redlich­Kwong equation with certain modifications. Since we wish to use our technique in a computer program as a subroutine for calculation of K factors at high pressures, we prefer to use

492 1& E C FUN 0 A MEN TAL S /

an analytical equation of state rather than the previously described corresponding-states method for estimating gas­phase properties (Gunn et at., 1966; Prausnitz and Gunn, 1958).

The Redlich-Kwong equation is now nearly 20 years old; recently it has been discussed by several authors (Barner tt al., 1966; Estes and Tully, 1967; Robinson and Jacoby, 1965; Wilson, 1964), and it is generally regarded as the best two­parameter equation now available (Shah and Thodos, 1965). For mixtures, however, it often gives poor results, as does the recent modification given by Redlich and coworkers (1965). The failure of the equation to give consistently g<?od results for mixtures is due to the inflexible mixing rules for the com­position dependence of the equation-of-state constants. We present here a modified mixing rule for the constant aj this modification incorporates one characteristic binary constant and we report such constants for 115 binary systems. Exten­sion to multicomponent systems follows without further assumptions and with no ternary (or higher) constants. A

.,

,

on

London,

1964 ). 966a).

\M ENTALS

European stcrdam,

)64.

t 15,1966 10,1967

TZ ,dey, Calif.

e 11 1.

1-

previously tating gas­, nd Cunn,

years old; arner el al., ;oby, 1965; e best two­dos, 1965). as docs the

,ers (1965) . ood results

or the eom-stants. \\'c tant a; this ry constant

s. Exten­out further nstants. A

I 1

1

I 1

I

j

somewhat similar study, restricted to paraffin-carbon dioxide mixtures, has recently been published by Joffe and Zudkevitch (1966). Other modifications of the Redlich-Kwong equation have been reported by Wilson (1964), Estes and Tully (1967), Robinson and Jacoby (1965), and Barner, Pigford, and Shreiner (1966).

F.ugocity CoeHicient

The fugacity of a component i in a gas mixture is related to the total pressure, P, and its mole fraction, Yb through the fugacity coefficient <Pt:

(1 )

The fugacity coefficient is a function of pressure, temperar ture, and gas composition; it is related to the volumetric properties of the gas mixture by either of the two exact rela­tions (Beattie, 1949; Prausnitz, 1959):

RT In <Pi = fP [(av) _ RTJ dP J 0 ont T,P,"f P .

(2)

RT In <Pt = - - - dV - RT In z }:'" [(oP) RTJ v ant T,V,"f V

(3)

where V is the total volume of the gas mixture, and z is the compressibility fjictor of the gas mixture at T and P. Since most equations of state are explicit in pressure, Equation 3 is more convenient to use.

For a mixture of ideal gases <Pi = 1 for all i. For a gas mix­ture that follows Amagat's assumption (Ot = vpuro t at the same T and P for the entire pressure range from zero to P), Equation 2 gives the Lewis fugacity rule (Lewis and Randall, 1923) which says

<PI = <ppuro I (at same T and P) (4)

This simplifying assumption, however, may lead to large error, especially for components present in sJl'llll concentra­tions. The Lewis fugacity rule becomes exact (at any pres­sure) only in the limitYt- 1.

The fugacity coefficient of component i in a gas mixture can be calculated from Equations 2 and 3 if sufficient volumetric data are available for the gas mixture. Since such data are not usually available, especially for multicomponent systems, fugacity coefficients are most often calculated by an extension of the theorem of corresponding states or with an equation of state. The method based on corresponding states has been discussed (Joffe, 1948; Leland et al., 1962). In the following we discuss a method based on an equation of state which is more convenient to use than the corresponding-states method.

Equation of State

Only the vi rial equation has a sound theoretical foundation for representing the properties of pure and mixed gases. When truncated after the third term, the vi rial equation is useful up to a density nearly corresponding to the critical density. A method for estimating the third virial coefficient of mixtures is given elsewhere (Chueh and Prausnitz, 1967a,b; Orent­licher and Prausnitz, 1967). For application at higher densi­ties, an empirical equation of state such as the Redlich-Kwong equation (1949) is more reliable. For vapor-phase fugacity coefficients we use the Redlich-Kwong equation throughout the entire range of density.

The 'Redlich-Kwong equation is

p ... RT

v - b

a

1'0.6 v(v + b)

/

(5)

where

a OoI(lT;/

(6) Pet

b = nbRTet

(7) Pet

The dimensionless constants no and nb are, respectively, 0.4278 and 0.0867 if the first and second isothermal derivatives of pressure with respect to volume are set equal to zero at the critical point. In vapor-liquid equilibria, however, we are interested in the volumetric behavior of saturated vapors over a relatively wide range of temperature, rather than in the critical region only. We propose, therefore, to evaluate no and nb for each pure component by fitting Equation 5 to the volumetric data of the saturated vapor. The temperature range used is that from the normal boiling point to the critical temperature. Table I lists 0 0 and nb for the saturated vapors of 19 pure substances most often encountered in high pressure vapor-liquid equilibria.

To apply Equation 5 to mixtures, we need a mixing rule. , We propose

(8)

where

(9)

and

N N a = ~ ~ Ji'YJflII (all;r6 v' aj(OJj) (10)

;_1 j_1

where

(11)

(12)

, (13)

Table I. Acentric Factors and Dimensionle" Constants In Redlich-Kwong Equation of State for Saturated Vapor'

w O. fit,

Methane 0.013 0.4278 0.0867 Nitrogen 0 .040 0 .4290 0 .0870 Ethylene 0 .085 0.4323 0 .0876 Hydrogen sulfide 0 . 100 0 .4340 0 .0882 Ethane 0.105 0 .4340 0.0880 Propylene 0.139 0.4370 0.0889 Propane 0.152 0.4380 0.0889 Isobutane 0.187 0.4420 0.0898 Acetylene 0.190 0 .4420 0.0902 1-Butene 0.190 0.4420 0 .0902 n-Butane 0.200 0.4450 0.0906 Cyclohexane 0 .209 0 .4440 0 .0903 Benzene 0.211 0.4450 0.0904 Isopentane 0.215 0 .4450 0 .0906 Carbon dioxide 0 .225 0 .4470 0.0911 n-Pentane 0.252 0 . 4510 0 .0919 n-Hexane 0.298 0 .4590 0.0935 .n-Heptane 0.349 0.4680 0 .0952 n-Octane 0 .398 0 . 4760 0.0968

VOL 6 NO.4 NOVEMBER 1967 493

I.

I I i I -I

Table II. Charaderlstlc Constant k12 for Binary Systems

[kit - 1- TOil ] ( Ton To .. )I"

System 2 kit X 10·

Methane Ethylene 1 Ethane 1 Propylene 2 Propane 2 n-Butane 4 Isobutane 4 n-Pentane 6 Isopentane 6 n-Hexane 8 Cyclohexane 8 n-Heptane 10 n-Octane (12)0 Benzene (8) Toluene (8) Naphthalene 14

Ethylene (or ethane) Ethane 0 Propylene 0 Propane 0 n-Butane 1 lsobutane 1 n-Pentane 2 lsopentane 2 n-Hexane 3 Cyclohexane 3 n-Heptane 4 n-Octane (5) Benzene 3 Toluene (3) Naphthalene 8

Propylene (or propane) Propane 0 n-Butane 0 Isobutane 0 n-Pentane 1 Isopentane 0 n-Hexane (1) Cyclohexane (1 ) n-Heptane (2) n-Octane (3) Benzene 2 Toluene (2)

n-Butane (or isobutane) Isobutane

" 0

n-Pentane 0 Isopentane 0 n-Hexane 0 Cyclohexane 0 n-Heptane 0 n-Octane (1 ) Benzene (1) Toluene (1 )

n-Pentane (or isopentane) Isopentane 0 n-Hexane 0 Cyclohexane 0 n-Heptane 0 n-Octane 0 Benzene (1 ) Toluene (1)

n-Hexane (or cyc1ohexane) n-Heptane 0 n-Qctane 0

o Numbers in parentheses are interpolated or estimated values.

1 V I/I - _ (v 1/3 + V 1/3)

ejJ - 2 Cj eJ

ZCf/ = 0.291 - 0 .08 (Wj ~ WJ)

TejJ = VTcjJ Tcjj (1 - kjJ)

(14)

(15)

(16)

The binary constant kjJ represents the deviation from the geometric mean for TejJ' It is a constant characteristic of the i-j interaction; to a good approximation, k'J is independent of the temperature, density, and composition. In general, k jJ must be obtained from some experimental information about the binary interaction. Good sources of this informa-

494 I&EC FUNDAMENTALS /

System 2 kll X 10'

Benzene (1) Toluene 1

n-Heptane n-Octane 0 Benzene (1 ) Toluene (1)

n-Octane Benzene (1) Toluene (1)

Benzene Toluene (0) Carbon dioxide Methane (5 ± 2)

Ethylene 6 Ethane 8 Propylene 10 Propane 11 ± 1 n-Butane 16 ± 2 Isobutane (16 ± 2) n-Pentane (18 ± 2) Isopentane (18 ± 2) Naphthalene 24

Hydrogen sulfide Methane 5 ± 1 Ethylene (5 ± 1) Ethane 6 Propylene (7) Propane 8 n-Butane (9) Isobutane (9) n-Pentane 11 ± 1 Isopentane (11 ± 1) Carbon di-

oxide 8 Acetylene Methane (5)

Ethylene 6 Ethane 8 Propylene 7 Propane 9 n-Butane (10) Isobutane (10) n-Pentane (11 ) Isopentane (11)

Nitrogen Methane 3 Ethylene 4 Ethane 5 Propylene (7) Propane (9) n-Butane 12 Helium 16

Argon Methane 2 Ethylene 3 Ethane 3 Oxygen 1 Nitrogen 0 Helium 5±1

Tetrafluoromethane Methane 7 Nitrogen 2 Helium (16 ± 2)

Hydrogen Methane 3 Neon Methane 28

Krypton 20 ± 2 Krypton Methane 1

tion are provided by second virial cross coefficients (Prausnitz, and Gunn, 1958) or by saturated liquid volumes of binary systems (Chueh and Prausnitz, 1967b), Table II presents our best estimates of kij for 115 binary systems. As new experi­mental data become available, this table should be revised and enlarged.

The proposed mixing rule for QH differs from Redlich's original mixing rule in two respects: (1) introduction of a binary constant k jil and (2) combination of critical volumes and compressibility factors to obtain a'j according to Equa­tions 12 through 15. As a result of (2), the proposed mixing rule does not reduce to Redlich's original rule even when

kif = 0, except when vec/veJ is close to unity; in general,

i .)

~ , ..

;..

( 102

1) 1 o 1 ) 1) 1) 1 ) 0) ± 2) 6 8 o ±1 ±2 ± 2) ± 2) ± 2) ~4 ±1 ± 1) 6

(7) 8

(9) (9) ±1 ± 1)

8 (5) 6 8 7 9

' 10) (10) {11) (11)

3 4 5

(7) (9 ) 12 16

2 3 3 1 o

5 ± 1 7 2

6 ± 2) 3

28 o ± 2

1

(Prausnitz, of binary

resents our ew experi­'evised and

\ Redlich's lction of a lal volumes g to Equa­,sed mixing even when in general,

j

Rcdlich's original rule gives a value for alj slightly smaller than that given by the proposed rule with klj = O. For example, lor the methane-n-pentane system, Redlich's original rule gives al) = 5.80 X 106 (p.s.i.a.) (0 R .)O.6 (cu. ft./lb. mole)2, whereas the proposcd rule gives 6.26 X 106 with klj = 0, and 5.70 X 106 with the recommended klj of 0.06. For this .particular case, Redlich's original rule gives a value of alj close to that of the proposed rule With the recommended klJ (as obtained from second virial cross coefficients) . On the other hand, for the system ethane-acetylene, for which v,/v'j is nearly unity, Redlich's original rule gives an alj which is essentially equal to that given by the proposed rule with kij = 0; such ajj is 12% too large as compared to the alj given by the proposed rule with a recommended k jj of 0.08. Finally, for the system carbon dioxide-n-butane, alj from Redlich's original rule corresponds to that of the proposed rule with kij = 0.04, as compared to a klJ of 0.18 obtained from second virial cross coefficients. This may explain why Redlich's original rule often works well for paraffin-paraffin systems but not for systems containing chemi­cally dissimilar components.

Fugacity Coefficients from Revised Redlich-Kwong Equation

By substituting Equation 5 and the mixing rules, Equations 8 to 16, into Equation 3, the fugacity coefficient of component k in the mixture becomes

N

V bl: 2 L Yjajk + b In 'Pk = In-- + __ i_I In V __ +

v - b v - b RP/2b v

abl: [ v + b b ] I Pv RTJ/2 b2 In -v- - v + b - n RT (17)

The molar volume, v, is that of the gas mixture; it is obtained by solving Equation 5 (which is cubic in v) and taking the largest rea l root for v. .-.

Figure 1 shows experimental and calculated compressibility factors for three mixtures of nitrogen and n-butane at 3100 F. (Evans and Watson, 1956). The values ofna and nb used are those for the pure saturated vapors, although the gas mixtures at this temperature are actually supercritical. The calculated compressibility factors for the gas mixture are in good agree­ment with the experimental data up to a pressure as high as 9000 p.s.i.a.

z

1.8r----r---r---r--r---,r---,r---..,r---..,---. -- CALCULATED WITH' k,2aO.12 --- CALCULATED WITH k,2 ' °

o DATA OF EVANS AND WATSON

• YH2 " 0.7016 b YH. -0.4974

L4 c YN • .:"",0,,-,. 2'-'!9'-'!S7!-,.. ____ +-_r~~~£...-l

0.6r----"''''''-'t-----+-----+-----+--~

o 2000 4000 6000 PRESSURE, pSia

8000

Figure 1. Compressibility factors of nitrogen-n-butane mixtures at 310 0 F.

leu - 0.12 obtained from second virial coeHicient data

I /

Figure 2 shows experimental and calculated fugacity co­efficients of carbon dioxide in a mixture containing 85 mole % n-butane at 3400 F. The experimental fugacity coefficients of carbon dioxide are obtained from the volumetric data of Olds et at. (1949) by performing graphically the differentiation and integration indicated in Equation 2. The test is a rather stringent one, since the mole fraction of carbon dioxide is small and the gas mixture is near its critical temperature. The fugacity coefficient of carbon dioxide shows an unusual pressure dependence, going through two inflections and a sharp maximum. The agreement is good considering the uncer­tainty involved in the numerical differentiation of the experi­mental data. Also indicated is the poor result obtained when the geometric mean assumption is used for T'l! ' The Lewis fugacity rule fails badly at all pressures, since the mole fraction of carbon dioxide is small .

Figure 3 shows experimental and calculated fugacity co­efficients of ethane in an equimolar mixture with methane. The experimental fugacity coefficients of ethane are those reported by Sage aqd Lacey (1950). Figures 4,5, and 6 show experimental and calculated compressibility factors for satu­rated vapor mixtures of three binary systems: propane­methane (Reamer et at., 1950), n-pentane-methane (Sage et at., 1942), and n-pentane-hydrogen sulfide (Reamer et at., 1953). Compressibility ' factors of the saturated vapors of the first two binary systems first increase with rising methane concentration but soon decrease because of the effect of high pressure. The calculated compressibility factors reproduce this behavior. The agreement is good except for the n-pentane­methane system near the critical.

For the n-pentane-hydrogen sulfide system at 1600 F., both components arc subcritical and the experimental and calculated compressibility factors are in good agreement over the entire composition range, from pure n-pentane to pure hydrogen sulfide. Calculations based on Redlich's original mixing rules (Redlich and Kwong, 1949) show less satisfactory agreement with experimental data. Also shown are the calculated fu­gacity coefficients for the heavy components; the fugacity co­efficients of propane and n-pentane at high pressure and low

1.8r---r---r----r---r-----r----. o FROM VOLUMETRIC DATA OF

OLDS, REAMER, SAGE 8 LACEY Q. CALCULATED WITH kIZ -O.18

b. CALCULATED WITH k,2 -O

LEWIS. FUGACITY RULE

14t--f-----.:l"t------.------1

Q6~-~-~~-~-~~-~_=~ o 2000 4000 6000 PRESSURE, psia

Figure 2. Fugacity coefficients of carbon dioxide in a mixture containing 85 mole % n-butane at 340 0 F;

leu - 0.18 obtained from second viriol coefficient data

v 0 I. 6 NO • .04 NOV E M B E R 1967 .0495

o DATA OF SAGE Ef. AL

CALCULATED WITH kIZaO.OI

0.81--~~-_+--_+_--+_--t_-_l

0.6f-----+-----"-..+-::.....--+----+---f--~

ck Ct,is 0.41---+--_+_-~'k--_+--~--c1c=i

021---+--+---+--~--~-~

O~_~~~~~~~ __ ~~~~~~ o 500 1000 1500 2000 2500 3(0) PRESSURE I psia

Figure 3. Fugacity coefficients of ethane in an equi­molar mixture with methane

leu = 0.01 obtained from second virial coefficient data

o VOLUt.4ETRIC DATA OF SAGE ET AL -- CALCULATED WITH klZ - 0.02

oJ I.OI----,-----,----r---~---+-~ I

;::1 ii:1 vI

0.41---+--+---~-_""__='---t--r-_;

O'21--_+---+---+---t_--t--,..----l

o~_~~_~~~~_~~_~~~~ o 250 500 750 1000 1250 1500 PRESSURE . psia

Figure 4. Compressibility factors and fugacity coef­ficients for saturated vapor of propane (1 )-methane (2) system at 100° F.

Ie,. = 0.02 obtained from saturated liquid volume data

concentration are extremely small. These results indicate very large deviations from ideal-gas behavior and from the Lewis fugacity rule; they point to the need for serious con­sideration of vapor-phase imperfections in high-pressure vapor­liquid equilibria.

Quantum Gases

The configurational properties of low-molecular-weight gases (hydrogen, helium, neon) are described by quantum, rather than classical, statistical mechanics. As a result, the properties of these gases cannot be given by the same corre­sponding-states treatment (Equations 9, and 11 to 16) as that

.496 I & E C FUN 0 A MEN TAL S /

o VOLUt.4ETRIC DATA OF SAGE ET AL -- CALCULATED WITH kIZ-O.06

--.,---....----+---+11.0 I I

-II ~I

I-~-_+_=-.;;:--_+_::_-=-:-:::_::__I_--_+_-- ~ 0.9 1-1 OCI VI

I

0.6t------''t----t---~..,..._-_t_~-_H: 0.8 I I Z I

0.41----1-~~--_+-___:;;_+.;:;:;:;~~0.7

O'21----+---+---....:.-~~-+_-_I_I

O=-_~~-~~~~~~~~~ o 500 1000 1500 2000 PRESSURE. psia

Figure 5. Compressibility factors and fugacity coef­ficients for saturated vapor of n-pentane (1 )-methane (2) system at 100° F. .

z

leu = 0.06 obtained from saturated liquid volume data

o VOLUMETRIC DATA OF REAMER ET AL CALCULATED WITH kIZ s O.12, CALCULATED WITH REDLICH S ORIGINAL

1.01t-----rM"'IX"'I.!.!N,..G....!.R'-""r ........ ---r-----.----i

PURE "-PENTANE

0.9Hr'-'"~r----t---t---t-----i

o.8~-~r--~~-

or O. 7t----p~--t---_t~c__-:-:::I:-:-::----I

'I H~

0.6t----+-----=rl-~-_+---+----l

O.5~-_;~---+-_T_-+~--_+---l

O.4~-~~-~;:---~~__::~___:~ o 200 800 1000

Figure 6. Compressibility factors and fugacity coef­ficients for saturated vapor of n-pentane (1)­hydrogen sulfide (2) at 160° F.

leu = 0.12 obtained from second virial coefficient data

used for classical gases when the true critical constants are used as the reducing parameters. It is possible, however, to define temperature-ciependent, effective critical constants (Gunn

. made to coincide with those for classical gases. These effective el at., 1966) with which the properties of quantum gases can be [

~-----------=----~~~

.0

').9

),8

z

D.7

Q6

0.5

:oef­one

IzS

000

:oef-(1)-

a

are used to define

(Gunn s can be effective

critical constants were found to depend on the molecular mass, Tn, and temperature in a simple manner. The effective critical temperature and effective critical pressure are given by

where

+~ mT

po P, = ---=-'-

+..:!.. mT

C1 = 21.8° K.

C2 = 44.2° K.

(18)

(19)

(20)

(21)

T, ° and Pc ° are, respectively, the classical critical tempera­ture and pressure-i.e ., the effective critical temperature and pressure in the limit of high temperature. Table III gives T, ° and P, ° for nine quantum gases.

For mixtures containing one or more of the quantum gases, Equations 8 through 12 are used, exeept that T" and pc/

are given by Equations 18 and 19; further, Tefl and Pcu are given by

where

VT;"T;ii (1 - k/i) +~

m/iT

P;u P,u = ---'-'--

+~ m'JT

;,i = ~ (~/ + ~)

(22)

(23)

(24)

(25)

(26)

(27)

For all quantum gases, Oa and Ob are, respectively, 0.4278 and 0.0867, and w(effective) is zero. Values of Vc ° for quan­tum gases are calculated from the relation v. o = 0.291RT. ° / P,o; they are listed in Table III.

Figure 7 shows experimental and calculated compressibility factors for the two quantum gases, hydrogen (Johnston and White, 1948) and helium (Mann, 1962) at cryogenic tempera­tures. Calculated results were obtained from the Redlich­Kwong equation using effective critical constants. Good agreement is obtained over the entire temperature and pres-

Table III. Ciani cal Critical Constants for Quantum Gases

v. o, Cc./

T~o, 0 K. p. o, Atm. Gram Mole

Ne 45.5 26.9 40 .3 He' 10 .47 6.67 37 .5 He' 10 .55 5.93 42.6 I-h 43.6 20.2 51.5 I-/D 42 .9 19 .6 52 .3 I-IT 42.3 19 .1 52 .9 D2 43 .6 20 .1 51.8 DT 43 .5 20.3 51.2 Ts 43.8 20.5 51.0

/

sure range, including the isotherms very close to the critical. The slightly high compressibility factors near the critical point are due to an inherent limitation of the Redlich-Kwong equa­tion which gives z, = 1/3 for all gases when Oa = 0.4278 and Ob = 0.0867.

Figure 8 shows experimental and calculated fugacity co-

1.2r--r--r---"ir---r--"--'7"""---'-"'7-I---, HELIUM

z

O.61--H1r7~'f-----f------1----I

CALCULATED o DATA OF MANN

z

-- CALCULATED o DATA OF JOHNSTON ANOWHITE

Figure 7. Compressibility foctors of helium and hy­drogen at low temperatures

-- CALCULATED WITH k,zoO.03

a 06 FROM SOLID-VAPOR EOUILIBRIUM DATA OF HIZA AND HERRING

--- CALCULATED WITH REDLICH'S ' ORIGINAL MIXING RULES

O.II---~-~r---""'::'''-

0.01 t-------1f-l~---+----.::::::::......d.

O·OQ,Ok----1--__ 5~O:;__-.L--7,:1 ObO----L-~ PRESSURE, ATMOSPHERES

Figure 8. Fugacity coefficients of methane in hydrogen at saturation

kit - 0.03 obtained from .econd viriol coefficient data

VOL 6 NO.4 NOVEMBER 1967 497

efficients of methane in a mixture with hydrogen at equilibrium with solid methane (Hiza and Herring, 1965). The good agreement at these low temperatures and high pressures sug­gests that the revised Redlich-Kwong equation can be success­fully applied to mixtures of nonpolar and quantum gases.

Conclusion

The equation of Redlich and Kwong provides a simple method for calculation of fugacity coefficients in nonpolar vapor mixtures at high pressures. The dimensionless con­stants fIa and fib which are universal in the original Redlich­Kwong equation are replaced by constants evaluated from the saturated volumetric properties of each pure component. This modification is relatively unimportant for the vapor phase, although it is vcry important (or thc liquid phase (Chueh and Prausnit1., 1967b). Thc large improvemcnt for vapor mix­tlln's is ohlained throllgh a modilication in thc mixing rille for the (·qllation.()f-~tat~· I'onslanl, {I,' for lIlixtlln:s, lhe lII:clIl'acy of the ('qllation is sigllilkantly inCl'easeu when lhe original, inflexible mixing rule is replaced by a flexible rule which con­tains one characteristic binary constant. Through ell'ective critical constants the equation of state can also be used for mixtures containing one or more of the quantum gases. Vapor­phase fugacity coefficients calculated by this modification of the Redlich-Kwong equation are useful for reduction and correlation of high-pressure phase equilibrium data (Chueh, 1967) .

Acknowledgment

The authors are grateful to the National Science Foundation for financial support and to the Computer Center, University of California, Berkeley, for the use of its facilities.

Nomenclature

a, b Cl, C2

/ k;j m

N

constants in Redlich-Kwong equation of state = constants given by Equations 20 and 21 = fugacity

characteristic constant for i-j interaction molecular weight molecular weight characteristic of i-j interaction;

the reduced mass number of components in the mixture number of moles of component i total pressure critical pressure

= classical critical pressure (high-temperature limit of effective critical pressure)

= critical pressure characteristic of i-j interaction = gas constant

temperature = critical temperature

classical critical temperature (high-temperature limit of effective critical temperature)

critical temperature characteristic of i-j interaction molar volume of vapor phase critical volume classical critical volume (v.o = O.291RT.o /p.o

for quantum gases) = critical volume characteristic of j-j interaction = mole fraction

498 I&EC FUNDAMENTALS / /

% = compressibility factor Z. %0 . = critical compressibility {actor

classical critical compressibility factor (0.291 for quantum gases)

cri.tical cC!mpressibility factor characteristic of i-j mteractlon

w = acentric factor 'Pi = fugacity coefficient of componerit i in a gas mixture fl., fib = dimensionless constants in Redlich-Kwong equation

SUPERSCRIPT

o = classical

SUBSCRIPTS

C = critical i, ii pure component I ij = i-j pair

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RECEIVED for review March 20, 1967 ACCEPTED June 5,1967

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